3.35.19 \(\int \frac {-2-\log (\frac {1}{x^2})}{32 x^2} \, dx\)

Optimal. Leaf size=11 \[ \frac {\log \left (\frac {1}{x^2}\right )}{32 x} \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {12, 2303} \begin {gather*} \frac {\log \left (\frac {1}{x^2}\right )}{32 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 - Log[x^(-2)])/(32*x^2),x]

[Out]

Log[x^(-2)]/(32*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2303

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*(d*x)^(m + 1)*Log[c*x^n])/(
d*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{32} \int \frac {-2-\log \left (\frac {1}{x^2}\right )}{x^2} \, dx\\ &=\frac {\log \left (\frac {1}{x^2}\right )}{32 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} \frac {\log \left (\frac {1}{x^2}\right )}{32 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - Log[x^(-2)])/(32*x^2),x]

[Out]

Log[x^(-2)]/(32*x)

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fricas [A]  time = 0.76, size = 9, normalized size = 0.82 \begin {gather*} \frac {\log \left (\frac {1}{x^{2}}\right )}{32 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(-log(1/x^2)-2)/x^2,x, algorithm="fricas")

[Out]

1/32*log(x^(-2))/x

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giac [A]  time = 0.17, size = 9, normalized size = 0.82 \begin {gather*} -\frac {\log \left (x^{2}\right )}{32 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(-log(1/x^2)-2)/x^2,x, algorithm="giac")

[Out]

-1/32*log(x^2)/x

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maple [A]  time = 0.07, size = 10, normalized size = 0.91

method	result	size
derivativedivides	\(\frac {\ln \left (\frac {1}{x^{2}}\right )}{32 x}\)	\(10\)
default	\(\frac {\ln \left (\frac {1}{x^{2}}\right )}{32 x}\)	\(10\)
norman	\(\frac {\ln \left (\frac {1}{x^{2}}\right )}{32 x}\)	\(10\)
risch	\(\frac {\ln \left (\frac {1}{x^{2}}\right )}{32 x}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/32*(-ln(1/x^2)-2)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/32*ln(1/x^2)/x

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maxima [A]  time = 0.45, size = 9, normalized size = 0.82 \begin {gather*} \frac {\log \left (\frac {1}{x^{2}}\right )}{32 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(-log(1/x^2)-2)/x^2,x, algorithm="maxima")

[Out]

1/32*log(x^(-2))/x

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mupad [B]  time = 2.06, size = 9, normalized size = 0.82 \begin {gather*} \frac {\ln \left (\frac {1}{x^2}\right )}{32\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(1/x^2)/32 + 1/16)/x^2,x)

[Out]

log(1/x^2)/(32*x)

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sympy [A]  time = 0.09, size = 8, normalized size = 0.73 \begin {gather*} \frac {\log {\left (\frac {1}{x^{2}} \right )}}{32 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(-ln(1/x**2)-2)/x**2,x)

[Out]

log(x**(-2))/(32*x)

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